Lewis-Riesenfeld phase-space invariant method (LRIM) is of the classic formalisms for solving a TDSE with time-dependent parameters (e.g., mass \(m(t)\) ) [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. The idea behind the LRIM is to construct a PSIO corresponding to the Hamiltonian \(\hat{H}(t)\) . Up to a time-dependent phase factor, the
eigenstates of the PSIO ( \(\hat{\mathcal {I}}\) ) will satisfy the TDSE corresponding to \(\hat{H}(t)\)  [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}.
LRIM for various classes of TDSE had been studied extensively in the literature. Such as the harmonic oscillators, a particle moving in a time-dependent electromagnetic field, and a particle moving in non-commutative (NC) space with time-dependent NC parameters have been solved with the help of LRIM [17]}, [18]}, [19]}, [20]}, [21]}.
M Maamache et al. constructed a class of Lewis-Riesenfeld invariant operators (LRIO) for a free particle with TDEM [14]}.
However, there is a gap in the literature on the general study of LRIO for an FP with TDEM. The present paper aims to fulfill this gap. The construction of the most general quadratic LRIO for an FP with TDEM is one of the motivations behind the present article.
Moreover, the restrictions on the
parameters of the LRIO are determined so that the diagonalization of \(\hat{\mathcal {I}}\)  with the help of a similarity transformation (symplectic group \(sp(2,\mathbb {R})\) ) is possible. It turns out that, in diagonal representation, \(\hat{\mathcal {I}}\)  can be factorized in terms of annihilation (\(\hat{a}\) ) and creation (\(\hat{a}^\dagger \) ) operators. The eigenstates along with the corresponding eigenvalues of \(\hat{\mathcal {I}}\)  are then determined from the eigenstates of \(\hat{a}\) . To obtain the complete solution of TDSE, we have computed the time-dependent phase factor (both the geometrical and dynamical phases). With the help of the uncertainty relation in \(\hat{x}\)  and \(\hat{p}\) , we have shown that the ground state is a squeezed coherent state (CS).
Being the minimum uncertainty state, a CS mostly resembles the classical states [23]}, [24]}, [25]}, [26]}. On the other hand, the Wigner quasiprobability distributions (WQD) provide a phase-space representation of QM. it is customary to study the Wigner quasiprobability distributions (WQD) for CS. WQD is generally felt to offer several advantages for use in modeling the behavior of physical processes. For instance, being a phase-space formulation, WQD involves both real space and the momentum space variables, distinctly different from SE [27]}, [28]}. It is believed that one can conceptually
identify where quantum corrections enter a problem by comparing it with the classical version. However, philosophical debates in this regard are inevitable, which we shall avoid in the present article. Rather, we shall indulge ourselves in the computation of WQD for our
problem, keeping in mind that, being entirely real, WQD simplifies both the calculation and the interpretation of results. For this reason, it was a natural choice for the simulation of quantum transport in devices such as the resonant tunneling diode [27]}, [28]}.
